"In the matter of physics the first lessons should contain nothing but what is experimental and interesting to see. A pretty experiment is in itself often more valuable than twenty formulae extracted from our minds."
~ Albert Einstein

Momentum and Energy

NOTE: There will be a full report on this week's lab. It is due the following week, when your lab is not in session. You are required to bring your report to the CAP building and drop it in the plastic bin outside the door to CAP 212 during the same time and weekday that your lab met the previous week! Please see the "Laboratory Report" and "Report Grading Rubric" links in the "Resources" menu to the right for help with writing the report. Print the "Report Scoring Sheet" and use it as your Title Page.

Introduction

In this lab we will be examining the conservation of momentum and energy using a collision between two carts.

In any real collision, energy is always lost. When scientists want to talk about the percentage of energy lost during a collision, they refer to the collision’s elasticity. An idealized collision where no energy is lost (imagine a ball that will bounce forever) would be called an elastic collision. There is never a real collision in which no energy is lost, just like there is no ball that exists that will bounce forever, but some collisions have such little energy loss that they are approximately elastic, just like some situations have such little air resistance or friciton that we assume there is none at all.

Collisions in which the total kinetic energy of the two bodies after the collision does not equal their total kinetic energy before the collisions are called inelastic collisions. A completely inelastic (or perfectly inelastic) collision is one in which the two colliding object move off together with the same velocity after the collision. Examples are a dropped ball that sticks to the floor rather than bounces or railroad cars that couple together when they collide. In inelastic collisions some of the kinetic energy is transformed into other forms of energy such as heat, noise, or stored elastic energy.

Momentum (p) is another quantity we could examine. Momentum is mass times velocity.

p = mv          (1)

During a collision, each individual object in the system may experience a change in momentum. However, if there are no net external forces acting on a system, then the total momentum will be conserved, no matter whether the collision is elastic or inelastic.

pbefore = pafter          (2)

A change in momentum is called an impulse (J).

J = Δp          (3)

Impulse is also the product of average force and time.

J = FΔt         (4)

When two carts collide, the elapsed time of the collision is the same for both. Newton's third law states that the carts will experience the same force, only in the opposite direction. therefore, Eq. (4) predicts that the carts should have the same impulse, only in the opposite direction. Therefore, the overall change in momentum, in the absence of external forces, will be zero.

Preliminary Questions

1. Think of three examples that can be approximated as totally elastic collisions.

2. Think of three examples of totally inelastic collisions.

3. Imagine you are dropping a ball. Is the momentum of the ball conserved as it falls? Explain your answer in terms of mass, velocity, and external forces.

4. Now think of the earth and the ball as a system as the ball falls toward the earth. Ignoring air friction, are there any other external forces other than the gravitational force between the earth and the ball? Is the momentum of the earth/ball system conserved? How must the Earth react to the ball in order for momentum to be conserved?

5. Next think a rear-end collision between a stationary car and a moving cars in which the cars stick together. Assume that air resistance and the rolling friction between the car tires and the road is small enough to ignore. Is energy conserved? Are the momenta of the individual cars conserved? What about the momentum of the two-car system?

Materials
Vernier dynamics track 
Two low-friction dynamics carts with magnetic and Velcro™ bumpers
Extra 200g masses for pan balance
Double pan balance
Cart masses                            
Motion detector
LabPro and computer with Logger Pro

Activity 1. Predicting the results of collisions


Figure 1: Carts on a track with two motion detectors.

Set two carts on a flat track such that the magnets on the ends are facing each other (see Figure 1). When you push the carts toward each other, providing they do not actually collide, the result should be a fairly elastic collision. On the opposite ends from the magnets are Velcro stickers. If the carts collide with these ends facing, the collision should be fairly inelastic.

Before actually observing collisions, let’s make some predictions.

For each of the following situations, predict how the momentum of each cart, and the momentum of the two-cart system, will change after the collision. The momentum of the two-cart system, also known as the total momenutm, is just the sum of the momentum of each of the two carts.

pCartA + pCartB   = ptotal        (5)

You can recreate the table below, but make the cells larger to give yourself plenty of room to write your predictions.

A table (example below) showing the direction of the momentum of each cart and the system comprised of both carts – both before and after the collision – is a useful way of displaying this information.  Just a simple  →,  0,  or  ←  is all that is needed (no numbers!).

If the cart is moving to the left, its momentum arrow will be pointing toward the left. If the cart is moving to the right, its momentum arrow will be pointing toward the right. If Cart B is carrying Mass C, then assume for your predictions that Cart B has an greatly-increased mass.

If the magnitudes are different, or if the momentum of either cart has increased or decreased after the collision, indicate this by changing the length of the momentum arrow. Assume that Cart A is on the left, and is always initially moving to the right.

Helpful hints: Do not cause violent collisions between carts! Keep the velocities low (gentle pushes). Also, position the mass on the cart so that it does not slide during the collision.

1a) A moving Cart A collides inelastically with a stationary Cart B.

1b) A moving Cart A collides elastically with a stationary Cart B.

2a) A moving Cart A collides inelastically with a stationary Cart B that is carrying Mass C.

2b) A moving Cart A collides elastically with a stationary Cart B that is carrying Mass C.

3a) A moving Cart A collides inelastically with Cart B that is moving towards it.

3b) A moving Cart A collides elastically with a Cart B that is moving towards it.

4a) A moving Cart A collides inelastically with a Cart B that is moving towards it carrying Mass C.

4b) A moving Cart A collides elastically with a Cart B that is moving towards it carrying Mass C.

Momentum of a two-cart collision
Before the Collision
After the Collision
Cart A
Cart B
Sum of both Carts
Cart A
Cart B
Sum of both Carts
1a)
1b)
2a)
2b)
3a)
3b)
4a)
4b)

Table 1: Momentum of a two-cart collision


Activity 2: Graphing momentum and energy

In this activity you are going to graph momentum and kinetic energy during a collision (since the track is level, there should be no change in potential energy). You will need the masses and velocities of Cart A, Cart B, and Mass C (Mass C can be the 1.00 kg mass that is already on the table).

Connect two motion detctors to Logger Pro, and set them so that one looks at Cart A and the other looks at Cart B. Because the detectors are facing opposite directions, you should reverse the direction of the detector that is looking at Cart B.

Open Logger Pro. Click "Collect" and give the carts a push. Identify motion detector A and motion detector B, then figure out and write down which corresponds to position 1, velocity 1, position 2, and velocity 2 in logger pro. This will help with trouble-shooting if something goes wrong. You will be converting these velocities into momentum and energy. If you are having problem getting data, please ask your instructor for help.

Next convert the velocity data into momentum and energy data.

First, set up three "User Parameters" that contain the masses of Cart A, Cart B, and Cart B with Mass C on it. To do this, go to Data > User Parameters, and click Add. Label and enter all three masses, including units.

Next, create calculated columns which calculate the momentum and kinetic energy of each cart for situation 1a). Instead of actual velocities, use the appropriate velocity column for Cart A or Cart B. Instead of actually entering the masses numerically into your equation, make use of the User Parameters that you just created. Make sure that everything is correctly labeled, with units.

Momentum of Cart A =
Momentum of Cart B =
Energy of Cart A =
Energy of Cart B =

Show these calculations in your notebook, and also in your report!

Also calculate the total momentum and the total energy, which would just be the sum of the individuals.

Total Momentum = Momentum of Cart A + Momentum of Cart B          (6)
Total Energy = Energy of Cart A + Energy of Cart B          (7)

Next you will change the position graph so that it will plot your momentums. Left-click on Y-axis of the upper graph, click "more", and set this graph to plot three calculated data sets: the momentum of each individual cart and the total momentum.

Next change the velocity graph so that it will plot your energies. Left-click on Y-axis of the lower graph, click "more", and set this graph to plot the remaining three calculated data sets: the energy of each individual cart and the total energy.

Now collect data for situation 1a), an inelastic (Velcro) collision between two carts. Are the results what you would expect? Can you identify each line on the graph as it corresponds to the momentum or energy of Cart A and Cart B? Can you identify the total momentum? Can you identify where on the graph the collision takes place?

Call your instructor over to verify that you are getting good data and are correctly interpreting the graphs.

Recreate Table 1 above, but add another column. This column will be percent difference between the total momentum before and after the collision. You will use Analyze > Interpolate on the momentum graph to find the momentum of each cart, and the sum of both, just before and just after the collision.

Make another table of energy data like Table 2 below.

 
Total KE before collision
Total KE after collision
% of Ke lost during the collision
1a)
1b)
2a)
2b)
3a)
3b)
4a)
4b)

Table 2: Kinetic Energy in collisions


Determine the total kinetic energy before the collision and the total energy after the collision using Analyze > Interpolate on the energy graph. Then calculate the percentage of kinetic energy lost during the collision (see Eq. (8) below) for the third column.

Percent energy lost = (KEbefore - KEafter) / KEbefore         (8)

Show example calculations in your report. You don't have to show every single similar calculation as long as you display the results in a table.

Print the graph of 1a) for your notebook. You will also print the graph of 1b), but not 2a) through 4b) (to save paper).

After you complete 1a), perform the other seven situations and fill in the tables.

IMPORTANT: Remember to update your calculated column equations when you add put Mass C onto Cart B! If you add Mass C to the wrong cart, or forget to factor it into your momentum equations, you will get incorrect results!

Discuss your results. Did they match your predictions? Why or why not?

Was momentum conserved in each collision? Explain.

Did the percent of energy lost depend on the type of collision? Explain.

Activity 3: Impulse

For each of the situations above, calculate the impulse (change in momentum) of each cart during the collision. You have already collected this data, so no new experimentation is needed. Put these calculations in a table like Table 3 below, using the correct units. Again, show example calculations.

According to Newton's Third Law, the impulse of the two carts should be equal in magnitude. Using percent difference, compare the impulse of each cart during the collisions.

Impulse (J) during collisions
Impulse of Cart A (units)
Impulse of Cart B (units)
% difference
1a)
1b)
2a)
2b)
3a)
3b)
4a)
4b)

Table 3: Impulse during collisions


Did Newton's Third Law hold true? Were there any external forces at work? Describe the effects of these external forces.

 

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